Techniques for the measurement of the phase of a radiation wave field have many applications in fundamental physics and as a basis for a number of measurement techniques involving various physical properties. Examples of applications of phase measurement techniques include the fields of x-ray imaging, electron microscopy, optical microscopy as well as optical tomography and x-ray phase tomography.
Phase is typically measured using interferometers of various types. The key feature of interferometry is the ability to quantitatively measure the phase of a wave field. Whilst interferometry based techniques retain significant importance it has been recognised that non-interferometric techniques may be used to provide phase information. A number of non-interferometric approaches involve attempting to solve a transport of intensity equation for a radiation wave field. This equation relates the irradiance and phase of a paraxial monochromatic wave to its longitudinal irradiance derivative and is described in M. R. Teague, “Deterministic Phase Retrieval: A Green's Function Solution” J. Opt. Soc. Am. 73 1434–1441 (1983). The article “Phase imaging by the transport of intensity equation” by N. Streibl, Opt. Comm. 49 6–10 (1984), describes an approach based on the transport of intensity equation by which phase structure can be rendered visible by the use of defocus and digital subtraction of intensity data obtained at various defocused distances. This approach only provides for phase visualisation and does not provide for the measurement of phase shift. Another approach based on solving the transport of intensity equation is disclosed in T. E. Gureyev, K. A. Nugent, D. Paganin and A. Roberts, “Rapid phase retrieval using a Fast Fourier transform”, Adaptive Optics, Volume 23, (1995) Optical Society of America Technical Digest Series, page 77–79 and T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation”, Opt. Comm., 133 339–346 (1997). This approach allows the phase of a light field to be recovered from two closely spaced intensity measurements when an illuminating beam has an arbitrary, but everywhere non zero intensity distribution limited by rectangular aperture. Whilst this approach can be used for non-uniform intensity distributions the extent of the non uniformity is limited and introduces significant computational complexity. Consequently this approach is not able to cope with non uniformities introduced by some sample absorption profiles or in some intensity illumination distributions. This approach is strictly also only applicable to coherent wave fields.
The article K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin and Z. Barnea “Quantitative phase imaging using hard X-rays” (1996) 77 Phys. Rev. Left. 2961–2964 is also based on a solution to the transport of intensity equation. Again the technique described cannot be applied to a non-uniform intensity distribution.
Other approaches based on a solution to the transport of intensity equation limited to a requirement of uniformity are described in T. E. Gureyev, K. A. Nugent, A. Roberts “Phase retrieval with the transport-of-intensity equation: matrix solution with the use of Zernike polynomials” J. Opt. Soc. Am. A Vol 12, 1932–1941 (1995) and T. E. Gureyev, A. Roberts and K. A. Nugent “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness”, J. Opt. Soc. Am. A Vol 12, 1942–1946 (1995).
A technique for recovery of phase in the case of non-uniform illumination is described in T. E. Gureyev and K. A. Nugent “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination”, J. Opt. Soc. Am. A Vol 13, 1670–1682 (1996). This approach is based on a method of orthogonal expansions and can be computationally complex in implementation. In many applications this complexity makes the technique impractical.